Efficient Exponential Tilting for Portfolio Credit Risk

This paper considers the problem of measuring the credit risk in portfolios of loans, bonds, and other instruments subject to possible default under multi-factor models. Due to the amount of the portfolio, the heterogeneous effect of obligors, and the phenomena that default events are rare and mutually dependent, it is difficult to calculate portfolio credit risk either by means of direct analysis or crude Monte Carlo under such models. To capture the extreme dependence among obligors, we provide an efficient simulation method for multi-factor models with a normal mixture copula that allows the multivariate defaults to have an asymmetric distribution, while most of the literature focuses on simulating one-dimensional cases. To this end, we first propose a general account of an importance sampling algorithm based on an unconventional exponential embedding, which is related to the classical sufficient statistic. Note that this innovative tilting device is more suitable for the multivariate normal mixture model than traditional one-parameter tilting methods and is of independent interest. Next, by utilizing a fast computational method for how the rare event occurs and the proposed importance sampling method, we provide an efficient simulation algorithm to estimate the probability that the portfolio incurs large losses under the normal mixture copula. Here the proposed simulation device is based on importance sampling for a joint probability other than the conditional probability used in previous studies. Theoretical investigations and simulation studies, which include an empirical example, are given to illustrate the method.